On the Relation between the $S$-matrix and the Spectrum of the Interior Laplacian
Volume 57 / 2009
Bulletin Polish Acad. Sci. Math. 57 (2009), 181-188
MSC: 78A45, 35J25.
DOI: 10.4064/ba57-2-11
Abstract
The main results of this paper are: 1) a proof that a necessary condition for $1$ to be an eigenvalue of the $S$-matrix is real analyticity of the boundary of the obstacle, 2) a short proof that if $1$ is an eigenvalue of the $S$-matrix, then $k^2$ is an eigenvalue of the Laplacian of the interior problem, and that in this case there exists a solution to the interior Dirichlet problem for the Laplacian, which admits an analytic continuation to the whole space $\mathbb R^3$ as an entire function.
